On the punctual and local ergodic theorem for nonpositive power bounded operators in $L^ p_ {\textbf {C}}[0,1],\;1<p<+\infty$
HTML articles powered by AMS MathViewer
- by I. Assani PDF
- Proc. Amer. Math. Soc. 96 (1986), 306-310 Request permission
Abstract:
We show in this note that there exists a function $f \in { \cap _1}_{ < p < + \infty }L_{\mathbf {C}}^p[0,1]$ and for each $p$ an isomorphism $T:L_{\mathbf {C}}^p \to L_{\mathbf {C}}^p$ such that ${\text {su}}{{\text {p}}_{n \in {\mathbf {Z}}}}\left \| {{T^n}} \right \| < + \infty$ and $T$ does not satisfy the punctual ergodic theorem. We give also an example of a one-parameter semigroup $({T_t},t \geqslant 0)$ of power bounded operators in each $L_{\mathbf {C}}^p(1 < p < + \infty )$ for which the assertion of the local ergodic theorem $((1/t)\smallint _0^t{T_s}fds$ converge almost everywhere as $t \to {0_ + }$ for all $f \in {L^p}$ fails to be true.References
- A. Ionescu Tulcea, Ergodic properties of isometries in $L^{p}$ spaces, $1<p<\infty$, Bull. Amer. Math. Soc. 70 (1964), 366–371. MR 206207, DOI 10.1090/S0002-9904-1964-11099-5
- R. V. Chacon and S. A. McGrath, Estimates of positive contractions, Pacific J. Math. 30 (1969), 609–620. MR 251190, DOI 10.2140/pjm.1969.30.609
- M. A. Akcoglu, A pointwise ergodic theorem in $L_{p}$-spaces, Canadian J. Math. 27 (1975), no. 5, 1075–1082. MR 396901, DOI 10.4153/CJM-1975-112-7
- Alberto de la Torre, A simple proof of the maximal ergodic theorem, Canadian J. Math. 28 (1976), no. 5, 1073–1075. MR 417819, DOI 10.4153/CJM-1976-106-8
- D. L. Burkholder, Semi-Gaussian subspaces, Trans. Amer. Math. Soc. 104 (1962), 123–131. MR 138986, DOI 10.1090/S0002-9947-1962-0138986-6
- M. A. Akcoglu and L. Sucheston, Remarks on dilations in $L_{p}$-spaces, Proc. Amer. Math. Soc. 53 (1975), no. 1, 80–82. MR 377558, DOI 10.1090/S0002-9939-1975-0377558-X
- Moshe Feder, On power-bounded operators and the pointwise ergodic property, Proc. Amer. Math. Soc. 83 (1981), no. 2, 349–353. MR 624929, DOI 10.1090/S0002-9939-1981-0624929-2 U. Krengel, Monograph in preparation.
- Mustafa A. Akcoglu and Ulrich Krengel, Two examples of local ergodic divergence, Israel J. Math. 33 (1979), no. 3-4, 225–230 (1980). A collection of invited papers on ergodic theory. MR 571531, DOI 10.1007/BF02762162
- P. L. Ul′janov, Divergent Fourier series, Uspehi Mat. Nauk 16 (1961), no. 3 (99), 61–142 (Russian). MR 0125398
- P. L. Ul′yanov, A. N. Kolmogorov and divergent Fourier series, Uspekhi Mat. Nauk 38 (1983), no. 4(232), 51–90 (Russian). MR 710115
- Mustafa A. Akcoglu, Pointwise ergodic theorems in $L_{p}$ spaces, Ergodic theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1978) Lecture Notes in Math., vol. 729, Springer, Berlin, 1979, pp. 13–15. MR 550405 I. Assani, Sur la convergence ponctuelle de quelques suites d’opérateurs (a paraître). —, Sur les opérateurs à puissances bornées et le théorème ergodique ponctuel dans ${L^p}[0,1]$ (a paraître).
- A. M. Olevskiĭ, Fourier series with respect to general orthogonal systems, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 86, Springer-Verlag, New York-Heidelberg, 1975. Translated from the Russian by B. P. Marshall and H. J. Christoffers. MR 0470599, DOI 10.1007/978-3-642-66056-6
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 96 (1986), 306-310
- MSC: Primary 47A35; Secondary 47D05
- DOI: https://doi.org/10.1090/S0002-9939-1986-0818463-9
- MathSciNet review: 818463